The representation of the collision operator introduced above can be used to
project it on a subspace spanned by a set of basis functions chosen to provide a sufficiently good sampling of the active region in momentum space (the blurred annulus pictured in Fig.REF ). This yields a finite-size matrix that can be diagonalized to find the excitation eigenmodes and their eigenvalues, giving the decay rates. We have found that, although this direct approach works, it is more convenient to use a somewhat
different approach to the problem, which employs the angular distribution of
quasiparticle scattering in the active region near the Fermi surface. The angular distribution, besides being directly linked to the individual excitation modes and their lifetimes, as discussed below, also allows to establish an interesting connection to the \((T/T_F)^2\log (T/T_F)\)  decay rates found from self-energy calculations[1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}.
